Minimal Finite Model of Wedge Sum of Spheres

In {Barmak(2007),Barmak(2011)}, Barmak extensively investigates the minimal finite models of the n-dimensional sphere Sⁿ for all n≥ 0 and establishes the minimal finite model of the wedge sum of unit circles i= 1ⁿ S¹. In this work, we demonstrate that the minimal finite model of the Möbius band coincides with the minimal finite model of the unit circle S¹. Furthermore, we establish that the minimal finite models of the spaces S² S¹, S² S² consist of only seven points, while the minimal finite models of the spaces S¹ S¹ S² and S² S² S¹ contain eight points. Additionally, we thoroughly discuss all the necessary homotopy groups and homology groups of the aforementioned spaces to provide a comprehensive and self-contained presentation in the paper.